Consistency when we want to find the distribution of sum of random variables following each one a distribution

Question

I want to clarify a point that disturbs me among different cases.

I am interested in formulate correctly in a general case when we know the distribution of different random variables and we want to calculate the distribution of the sum of these random variables.

For example, the distribution of 2 random variables following a different uniform distribution is not the sum of the 2 distributions, we are ok ? (by the way, how to compute the distribution of this sum ?).

On another side, when we take 2 random variables following a different normal distribution, the PDF of the sum of both is a PDF which is the sum of 2 Gaussians ? I think that it would logical but I am not sure (regarding my first example above), especially for the normalization of the PDF of sum which has to be equal to 1 when integrate over all the domain.

Now, in my case, I have to compute the distribution of the following quantity :

$$\sum_{\ell=\ell_{\min }}^{\ell_{\max }} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}$$

with the random variable $a_{\ell m}$ following a normal centered on 0 and with a variance equal to $C_{\ell}$.

So, I decided to begin firstly by the quantity $\sum_{m=-\ell}^{\ell} a_{\ell m}^{2}$ from a distribution point of view :

We recall the properties of a few basic distributions.

1. $\mathcal{N}(0, C_{\ell})^2$ distribution is equivalent to $C_{\ell}\,\chi^2(1)=\Gamma(\frac{1}{2}, 2C_{\ell})$ distribution.
2. The distribution $\sum_{i=1}^N\Gamma(k_i, \theta)$ is equivalent to a $\Gamma (\sum_{i=1}^N k_i, \theta)$ distribution for independent summands.

Let us formulate the distribution followed by this random variable. Using previous points 1 and 2, we obtain : $$ \begin{align} \sum_{m=-\ell}^\ell (a_{\ell m})^2&= \sum_{m=-\ell}^{\ell} C_{\ell} \cdot\left(\frac{a_{\ell, m}}{\sqrt{C_{\ell}}}\right)^{2} \label{sum_alm} \end{align} $$

We can develop the distribution of this observable like this :

3. $a_{\ell m}$ follows a $\mathcal{N}(0, C_{\ell})$ distribution.
4. $\sum_{m=-\ell}^{\ell} C_{\ell} \cdot\left(\dfrac{a_{\ell, m}}{\sqrt{C_{\ell}}}\right)^{2}$ follows a $\sum_{m=-\ell}^{\ell} C_{\ell} \, \mathrm{\chi^2}(1)$ distribution.
5. $\sum_{m=-\ell}^{\ell} C_{\ell}\,\mathrm{\chi^2}(1)$ distribution is equivalent to a $C_{\ell}\,\sum_{m=-\ell}^{\ell}\, \mathrm{\chi^2(1)}$ distribution.
6. $C_{\ell} \sum_{m=-\ell}^{\ell} \mathrm{\chi^2}(1)$ is equivalent to a $C_{\ell} \,\mathrm{\chi^2}(2\ell+1)$ distribution.
7. $C_{\ell}\,\mathrm{\chi^2}(2 \ell+1)$ distribution is equivalent to $C_{\ell}\,\mathrm{Gamma}((2\ell+1)/2, 2)$ distribution.
8. $C_{\ell}\,\mathrm{Gamma}((2\ell+1)/2, 2)$ is equivalent to a $\mathrm{Gamma}((2\ell+1)/2, 2C_\ell)$ distribution.

We have taken the convention (shape,scale) parameters for $\mathrm{Gamma}$ distribution. Given the fact that we consider the random variable :

$$ \begin{equation} \sum\limits_{\ell=\ell_{min}}^{\ell_{max}}\sum\limits_{m=-\ell}^{\ell} a_{\ell m}^2 \end{equation} $$

This sum of random variables $\sum\limits_{m=-\ell}^{\ell} a_{\ell m}^2$ follows a Moschopoulos distribution : it represents the distribution of the sum of random variables each one following a $Gamma$ distribution with different shape and scale parameters.

Is this reasoning correct ? I mean, about the sum of random variables $\sum\limits_{m=-\ell}^{\ell} a_{\ell m}^2$ which follows a Moschopoulos distribution ?

Indeed, if it is correct, the random variables $\sum\limits_{m=-\ell}^{\ell} a_{\ell m}^2$ follow a $\text{Gamma}$ distribution with different shape and scale parameters.

As conclusion, my main questions :

1) Is my reasoning above correct ? I mean does the random variable $\sum_{\ell=\ell_{\min }}^{\ell_{\max }} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}$ follows a Moschopoulos distribution ?

2) How to express correctly the things when I want to say for example, the distribution of sum of random variable "$X_i$" is equivalent to a "given" distribution or the sum of random variables "$X_i$" follows a "given" distribution ?

3) How to demonstrate that the convolution operation preserve the area of convolution to be equal to 1 when we integrate from all the domain ?

I make confusions between "equality from a distribution point of view", "the following distribution of a random variable" and the "sum of PDF".

Hoping have been enough clear.

Answer 1

1) Is my reasoning above correct ? I mean does the random variable $\sum_{\ell=\ell_{\min }}^{\ell_{\max }} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}$ follows a Moschopoulos distribution ?

The reasoning is correct. You have a sum of variables $a_{\ell m}^2$ that are gamma distributed. That matches the description for the Moschopoulos distribution.

See also this question Generic sum of Gamma random variables

2) How to express correctly the things when I want to say for example, the distribution of sum of random variable "$X_i$" is equivalent to a "given" distribution or the sum of random variables "$X_i$" follows a "given" distribution ?

For example: "The distribution of the sum $ \sum\limits_{\ell=\ell_{min}}^{\ell_{max}}\sum\limits_{m=-\ell}^{\ell} a_{\ell m}^2$ follows a Moschopoulos distribution."

3) How to demonstrate that the convolution operation preserve the area of convolution to be equal to 1 when we integrate from all the domain ?

This is a property of convolution. If the convolution is

$$h(x) = \int_{-\infty}^\infty f(y)g(x-y)dy$$

Then the integral of $h(x)$ is

$$ \int_{-\infty}^\infty h(x) dx = \left( \int_{-\infty}^\infty f(x)dx \right) \cdot \left(\int_{-\infty}^\infty g(x)dx\right) = 1 \cdot 1 = 1$$

Which follows from Fubini's theorem which states that you can change the order of integration

$$ \begin{array}{} \int_{-\infty}^\infty h(x) dx & =& \int_{-\infty}^\infty \left(\int_{-\infty}^\infty f(y)g(x-y)dy\right)dx\\ &=& \int_{-\infty}^\infty \left(\int_{-\infty}^\infty f(y)g(x-y)dx\right)dy \\&=& \int_{-\infty}^\infty f(y)\underbrace{ \left(\int_{-\infty}^\infty g(x-y)dx\right)}_{=1}dy \\ &=& \int_{-\infty}^\infty f(y) dy \\ &=& 1 \end{array}$$